On sums of sets of primes with positive relative density

Abstract

In this paper we show that if A is a subset of the primes with positive relative density δ, then A + A must have positive upper density C1δefalse(normallogfalse(1/δfalse)false)2/3(log log(1/δ))1/3 in the natural numbers. Our argument uses the techniques developed by Green and Green‐Tao in their work on arithmetic progressions in the primes, in combination with a result on sums of subsets of the multiplicative subgroup of the integers modulo m.

Sums of positive density subsets of the primes

“…In [3] and [14] similar results as Theorem 1.3 appear with essentially same densities as in respective results for the primes. Proofs of those results in [3] and [14] proceed in different lines from ours.…”

“…Indeed Ramaré and Ruzsa [14] had proved Theorem 1.1 with (1 − o(1)) replaced by (c − o(1)) for some constant c. Until recently the authors of [3] as well as the current author were unaware of that work. In light of [14], the achievement of Theorem 1.1 is getting the right constant.…”

“…On the other hand Chipeniuk and Hamel [3] pointed out Example 1.2 and anticipated that the bound there is the right answer in the spirit of Freiman's theorem. Theorem 1.1 shows that this is indeed the case (It is clear from Theorem 2.1 below that in case A = B lower densities can be replaced by upper densities also in Theorem 1.1).…”

“…Chipeniuk and Hamel [3] proved essentially the following theorem, again using an adaption of methods of Green and Green-Tao together with arguments which show that there cannot be too bad obstructions like that in Example 1.2.…”

On sums of sparse prime subsets

“…Using 'W-trick', the strategy developed by Green-Tao, Chipeniuk and Hamel [5] showed that if A is a subset of the primes with positive relative lower density δ, then the set A + A has positive lower density at least…”